A stained glass window for the entrance of a new office building is to be designed in the shape of a rectangle?
August 19, 2011 by refl
Filed under Stained glass
capped with a semi-circle. The perimeter of the window is not to exceed 30 m. The price of the stained glass is $9.75/m^2. Determine the total cost of the glass required for this window to have a maximum area.
can u please explain what the variables stand for and what equation you are using. Thanks a lot
Let x be the width and y be the length of the rectangle.
x/2 is the radius of the semicircle
Perimeter of the Norman window is x+2y+(π x)/2
Let P be the perimeter
P = x+2y+(π x)/2——–(1)
Solving for y from equation (1)
2y = P-x-πx/2
y = P/2-x/2-πx/4——–(2)
Area = xy + π x^2 / 8
A = x(P/2-x/2-π x/4) + π x^2/8
A= Px/2-x^2 /2 -πx^2/4 +πx^2/8
dA/dx = P/2 -2x/2-2πx /4 +2πx / 8 =0
(4p-8x-2πx)/8=0
4p-2x(π+4)=0
4p=2x(π+4)
x= 2P / (4+π)
using equation (2)
y=P/2-P/(4+π)-2πP/4(4+π)
2(4+π)P-4P-2πP/4(4+π)
=4P/4(π+4) = P/(π+4)
I have used P for the perimeter. In the last line P=30.
Width of the window = 2(30) / (4+π)= 60 / (4+π) =x
Length of the window = 30/(π+4) = y
Area = xy + π x^2 / 8
Compute the area and multiply by 9.75 to get the total cost.
Note:
d^2A/dx^2 =-1-π/2+π/4 < 0, indicates that the area is maximized.
.
30=2r+2h+pi*r/2 The rectangle must be complete to sustain the semicircle
S= r*h +1/8pi r^2
h= (30-2r-pir/2)/2
S= 1/2*(30r-2r^2-1/2pi*r^2) +1/8pi*r^2
S¨=15-2r-pir^/2+1/4 pir = 15-r(2+pi/2-pi/4)=0 so r= 15/(2+pi/4)
=5.3852
h=5.3852
Cost = $393.80